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Find Conditions for Stationarity and Invertibility of Time Series Processes

Find Conditions for Stationarity and Invertibility of Time Series Processes

    
Find conditions for an ARMAProcess[2,2] to be weakly stationary, using WeakStationarity:
Wolfram Language code: arma = ARMAProcess[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, σ]; ws = WeakStationarity[arma]
Wolfram Language code: RegionPlot[ws, {Subscript[a, 1], -2, 2}, {Subscript[a, 2], -2, 2}, GridLines -> Automatic]
Find an instance of a weakly stationary ARMAProcess[2,2]:
Wolfram Language code: aspt = ProcessParameterAssumptions[arma]; rest = Subscript[a, 1] ≠ 0 && Subscript[a, 2] ≠ 0 && Subscript[b, 1] ≠ 0 && Subscript[b, 2] ≠ 0 && σ ≠ 0; wsARMA = arma /. FindInstance[ws && rest && aspt, {Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], σ}][[1]]
Check that the process is weakly stationary:
Wolfram Language code: WeakStationarity[wsARMA]
Find conditions for an ARMAProcess[2,2] to be invertible, using TimeSeriesInvertibility:
Wolfram Language code: in = TimeSeriesInvertibility[ARMAProcess[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, σ^2]]
Wolfram Language code: RegionPlot[in, {Subscript[b, 1], -2, 2}, {Subscript[b, 2], -2, 2}, GridLines -> Automatic]
Find an instance of an invertible ARMAProcess[2,2]:
Wolfram Language code: aspt = ProcessParameterAssumptions[arma]; rest = Subscript[a, 1] ≠ 0 && Subscript[a, 2] ≠ 0 && Subscript[b, 1] ≠ 0 && Subscript[b, 2] ≠ 0 && σ ≠ 0; invARMA = arma /. FindInstance[in && rest && aspt, {Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], σ}][[1]]
Check that the process is invertible:
Wolfram Language code: TimeSeriesInvertibility[invARMA]